Predictive properties and minimaxity of Bayesian predictive synthesis

We examine and compare the predictive properties of classes of ensemble methods, including the recently developed Bayesian predictive synthesis (BPS). We develop a novel strategy based on stochastic processes, where the predictive processes are expressed as stochastic differential equations, evaluated using It\^{o}'s lemma. Using this strategy, we identify two main classes of ensemble methods: linear combination and non-linear synthesis, and show that a subclass of BPS is the latter. With regard to expected squared forecast error, we identify the conditions and mechanism for which non-linear synthesis improves over linear combinations; conditions that are commonly met in real world applications. We further show that a specific form of non-linear BPS (as in McAlinn and West, 2019) produces exact minimax predictive distributions for Kullback-Leibler risk and, under certain conditions, quadratic risk. A finite sample simulation study is presented to illustrate our results.